IJMMS 30:1 (2002) 9–14
PII. S016117120201253X
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
THE EQUIVALENCE BETWEEN THE HAMILTONIAN
AND LAGRANGIAN FORMULATIONS
FOR THE PARAMETRIZATIONINVARIANT THEORIES
S. I. MUSLIH
Received 27 March 2001 and in revised form 13 August 2001
The link between the treatment of singular Lagrangians as field systems and the canonical
Hamiltonian approach is studied. It is shown that the singular Lagrangians as field systems are always in exact agreement with the canonical approach for the parametrization
invariant theories.
2000 Mathematics Subject Classification: 37K10, 37K05, 17B80.
1. Introduction. [2, 3, 4, 5], the Hamilton-Jacobi formulation of constrained systems has been studied. This formulation leads us to obtain the set of Hamilton-Jacobi
partial differential equations (HJPDE) as follows:
∂S ∂S
,
= 0,
Hα tβ , qa ,
∂qa ∂tα
α, β = 0, n − r + 1, . . . , n; a = 1, . . . , n − r ,
(1.1)
where
Hα = Hα tβ , qa , pa + pα ,
(1.2)
and H0 is defined as
H0 = pa wa + pµ q̇µ |pν =−Hν − L t, qi , q̇ν , q̇a = wa ,
µ, ν = n − r + 1, . . . , n.
(1.3)
The equations of motion are obtained as total differential equations in many variables as follows:
dqa =
dz =
∂Hα
dtα ,
∂pa
dpa = −
− Hα + pa
∂Hα
dtα ;
∂pa
∂Hα
dtα ,
∂qa
dpβ = −
∂Hα
dtα ,
∂tβ
(1.4)
α, β = 0, n − r + 1, . . . , n, a = 1, . . . , n − r ,
where z = S(tα ; qa ). The set of equations (1.4) is integrable (see [4, 5]) if
dH0 = 0,
dHµ = 0,
µ = n − r + 1, . . . , n.
(1.5)
If conditions (1.5) are not satisfied identically, we consider them as new constraints
and again test the consistency conditions. Hence, the canonical formulation leads to
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S. I. MUSLIH
obtain the set of canonical phase space coordinates qa and pa as functions of tα ; besides, the canonical action integral is obtained in terms of the canonical coordinates.
The Hamiltonians Hα are considered as the infinitesimal generators of canonical transformations given by parameters tα , respectively.
In [1], the singular Lagrangians are treated as field systems. The Euler-Lagrange
equations of singular systems are proposed in the form
∂L
∂qa
∂L
∂
−
= 0, ∂α qa =
,
(1.6)
∂tα ∂ ∂α qa
∂qa
∂tα
with constraints
dG0 = −
∂L
dt,
∂t
(1.7)
dGµ = −
∂L
dt,
∂qµ
(1.8)
where
L tα , ∂α qa , q˙µ , qa = L qa , qα , q˙a = ∂α qa t˙α ,
∂L
Gα = Hα qa , tβ , pa =
.
∂ q˙a
q˙µ =
dqµ
,
dt
(1.9)
In order to have a consistent theory, we should consider the variations of the constraints (1.7) and (1.8).
In this paper, we study the link between the treatment of singular Lagrangians as
field systems and the canonical formalism for the parametrization invariant theories.
2. Parametrization-invariant theories as singular systems. In [4], the canonical
method treatment of the parametrization-invariant theories is studied and will be
briefly reviewed here.
Consider a system with the action integral as
(2.1)
S qi = dt ᏸ qi , q̇i , t , i = 1, . . . , n,
where ᏸ is a regular Lagrangian with Hessian n. Parametrize the time t → τ(t), with
τ̇ = dτ/dt > 0. The velocities q̇i may be expressed as
q̇i = qi τ̇,
(2.2)
dqi
.
dτ
(2.3)
where qi are defined as
qi =
Denote t = q0 and qµ = (q0 , qi ), µ = 0, 1, . . . , n, then the action integral (2.1) may be
written as
qi
,
(2.4)
S qµ = dτ ṫ ᏸ qµ ,
ṫ
THE EQUIVALENCE BETWEEN THE HAMILTONIAN AND LAGRANGIAN . . .
11
which is parametrization invariant since L is homogeneous of first degree in the ve
locities qµ with L given as
qi
.
L qµ , q˙µ = ṫ ᏸ qµ ,
ṫ
(2.5)
The Lagrangian L is now singular since its Hessian is n.
The canonical method in [2, 3, 4, 5] leads us to obtain the set of Hamilton-Jacobi
partial differential equations as follows:
H 0 = pτ − L q0 , qi , q˙0 , q̇i = wi
+ piτ qi + pt q˙0 |pt =−Ht = 0,
H t = pt + Ht = 0,
pt =
pτ =
∂S
,
∂τ
(2.6)
∂S
,
∂t
where Ht is defined as
Ht = −ᏸ qi , wi + piτ wi .
(2.7)
Here, piτ and pt are the generalized momenta conjugated to the generalized coordinates qi and t, respectively.
The equations of motion are obtained as total differential equations in many variables as follows:
dqi =
∂H 0
∂H t
∂H t
dτ +
dq0 =
dq0 ,
∂pi
∂pi
∂pi
(2.8)
dp i = −
∂H 0
∂H t
∂H t
dτ −
dq0 = −
dq0 ,
∂qi
∂qi
∂qi
dpt = −
∂H 0
∂H t
dτ −
dq0 = 0.
∂q0
∂q0
(2.10)
dH t = dpt + Ht
(2.11)
(2.9)
Since
vanishes identically, this system is integrable and the canonical phase space coordinates qi and pi are obtained in terms of the time (q0 = t).
Now, we look at the Lagrangian (2.5) as a field system. Since the rank of the Hessian
matrix is n, this Lagrangian can be treated as a field system in the form
qi = qi (τ, t),
(2.12)
thus, the expression
qi =
∂qi ∂qi
ṫ,
+
∂τ
∂t
can be substituted in (2.5) to obtain the modified Lagrangian L :
1 ∂qi ∂qi
ṫ .
L = ṫ ᏸ qµ ,
+
∂t
ṫ ∂τ
(2.13)
(2.14)
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S. I. MUSLIH
Making use of (1.6), we have
∂
∂L
−
∂qi ∂t
∂L
∂
∂L
−
= 0.
∂ ∂qi /∂t
∂τ ∂ ∂qi /∂τ
Calculations show that (2.15) leads to a well-known Lagrangian equation as
∂ᏸ
d
∂ᏸ
= 0.
−
∂qi dt ∂ dqi /dt
(2.15)
(2.16)
Using (2.7), we have
Ht = −ᏸ +
∂ᏸ
q̇i .
∂ q̇i
(2.17)
In order to have a consistent theory, we should consider the total variation of Ht .
In fact,
∂ᏸ
dt.
∂t
(2.18)
∂L
dτ.
∂t
(2.19)
dHt = −
Making use of (1.8), we find that
dHt = −
Besides, the quantity H0 is identically satisfied and does not lead to constraints.
We notice that (2.8) and (2.9) are equivalent to (2.15) and (2.16).
3. Classical fields as constrained systems. In the following sections, we study the
Hamiltonian and Lagrangian formulations for classical field systems and demonstrate
the equivalence between these two formulations for the reparametrization-invariant
fields.
t) in four space-time dimensions may be
A classical relativistic field φi = φi (x,
described as the action functional
(3.1)
S φi = dt d3 x ᏸ φi , ∂µ φi , µ = 0, 1, 2, 3; i = 1, 2, . . . , n,
which leads to the Euler-Lagrange equations of motion as
∂ᏸ
∂ᏸ
= 0.
− ∂µ
∂φi
∂ ∂ µ φi
(3.2)
We can go over from the Lagrangian description to the Hamiltonian description by
using the definition
πi =
∂ᏸ
∂ φ̇i
,
(3.3)
then the canonical Hamiltonian is defined as
H0 = d3 x πi φ̇i − ᏸ .
(3.4)
The equations of motion are obtained
π̇i = −
∂H0
,
∂φi
φ̇i =
∂H0
.
∂πi
(3.5)
THE EQUIVALENCE BETWEEN THE HAMILTONIAN AND LAGRANGIAN . . .
13
4. Reparametrization-invariant fields. In analogy with the finite dimensional systems, we introduce the reparametrization-invariant action for the field system:
(4.1)
S = dτ ᏸR d3 x,
where
ᏸR = ṫ ᏸ φi , ∂µ φi .
(4.2)
Following the canonical method [2, 3, 4, 5], we obtain the set of [HJPDE],
(τ) dφi
H 0 = πτ + πi
H
t
dτ
= πt + Ht = 0,
+ πt
dt
− ᏸR = 0,
dτ
πτ =
∂S
,
∂τ
∂S
,
πt =
∂t
(4.3)
where Ht is defined as
(τ) dφi
,
Ht = −ᏸ φi , ∂µ φi + πi
dt
(4.4)
(τ)
and πi , πt are the generalized momenta conjugated to the generalized coordinates
φi and t, respectively.
The equations of motion are obtained as follows:
∂H 0
∂H t
∂H t
dτ +
dt =
dt,
∂πi
∂πi
∂πi
∂H 0
∂H t
∂H t
dτ −
dt = −
dt,
dπi = −
∂φi
∂φi
∂φi
dφi =
dπt = −
∂H t
∂H 0
dτ −
dt = 0.
∂t
∂t
Now the Euler-Lagrangian equation for the field system reads as
∂ᏸ
∂ᏸ
∂
= 0.
−
∂φi ∂x µ ∂ ∂φi /∂xµ
(4.5)
(4.6)
(4.7)
(4.8)
Again as for the finite-dimensional systems, (4.5) and (4.6) are equivalent to (4.8)
for field systems.
5. Conclusion. As it is mentioned in the introduction, if the rank of the Hessian
matrix for discrete systems is (n − r ), 0 < r < n, then the systems can be treated
as field systems [1]. The treatment of Lagrangians as field systems is always in exact
agreement with the Hamilton-Jacobi treatment for reparametrization-invariant theories. The equations of motion (2.8) and (2.9) are equivalent to the equations of motion
(2.15) and (2.16). Besides, the variations of constraints (2.18) and (2.19) are identically
satisfied and no further constraints arise.
In analogy with the finite-dimensional systems, it is observed that the Lagrangian
and the Hamilton-Jacobi treatments for the reparametrization-invariant fields are in
exact agreement.
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S. I. MUSLIH
References
[1]
[2]
[3]
[4]
[5]
N. I. Farahat and Y. Güler, Singular Lagrangians as field systems, Phys. Rev. A (3) 51 (1995),
no. 1, 68–72.
Y. Güler, Canonical formulation of singular systems, Nuovo Cimento B (11) 107 (1992),
no. 12, 1389–1395.
, Integration of singular systems, Nuovo Cimento B (11) 107 (1992), no. 10, 1143–
1149.
S. I. Muslih and Y. Güler, On the integrability conditions of constrained systems, Nuovo
Cimento B (11) 110 (1995), no. 3, 307–315.
, Is gauge fixing of constrained systems necessary? Nuovo Cimento B 113 (1998),
277–289.
S. I. Muslih: Department of Physics, Al-Azhar University, Gaza, Palestine
E-mail address: sami_muslih@hotmail.com